Multiple-pathways light modulation in Pleurosigma strigosum bi-raphid diatom

Ordered, quasi-ordered, and even disordered nanostructures can be identified as constituent components of several protists, plants and animals, making possible an efficient manipulation of light for intra- and inter- species communication, camouflage, or for the enhancement of primary production. Diatoms are ubiquitous unicellular microalgae inhabiting all the aquatic environments on Earth. They developed, through tens of millions of years of evolution, ultrastructured silica cell walls, the frustules, able to handle optical radiation through multiple diffractive, refractive, and wave-guiding processes, possibly at the basis of their high photosynthetic efficiency. In this study, we employed a range of imaging, spectroscopic and numerical techniques (including transmission imaging, digital holography, photoluminescence spectroscopy, and numerical simulations based on wide-angle beam propagation method) to identify and describe different mechanisms by which Pleurosigma strigosum frustules can modulate optical radiation of different spectral content. Finally, we correlated the optical response of the frustule to the interaction with light in living, individual cells within their aquatic environment following various irradiation treatments. The obtained results demonstrate the favorable transmission of photosynthetic active radiation inside the cell compared to potentially detrimental ultraviolet radiation.

1 Wide-angle beam propagation method (WA-BPM) Beam propagation method (BPM) is a computational tool routinely employed to simulate light propagation in waveguides and optical fibers, i.e. in paraxiality conditions (small angles with respect to the optical axis) and when a uniform refractive index along the direction of propagation of the field is considered.Since, in the case of diatom valves, the nano-porous structure induces diffraction and, after transmission through silica, the radiation propagates in a medium characterized by a different refractive index, it is mandatory to make use of a non-paraxial approximation of BPM.
Starting from the Helmoltz equation for a scalar field (i.e.neglecting polarization effects): with E(r, t) = ϕ(x, y, z)e −iωt scalar electric field, k = nk 0 wavenumber (with k 0 = 2π λ wavenumber in free space), and n = n(x, y, z) refractive index spatial distribution, we can write the solution as: i.e. the electric field can be expressed as the product of a slowly varying envelope factor U (x, y, z) and a rapid varying phase factor e −ikrz , where the reference wavenumber k r takes into account the average phase variation of the field.We are assuming that the considered wave propagates primarily along z (i.e.we are considering, at first, paraxial conditions).We will also suppose, for now, that the amplitude varies slowly along z axis too.Inserting U (x, y, z)e −ikrz into Eq.S1 we obtain: Making use of the slowly varying envelope approximation: we obtain the basic BPM equation: Specifying U (x, y, z) at a plane z = z 0 , we can iterate U along the z-axis using finite differences for the x and y derivatives.
A BPM variant which can take into account non-paraxial conditions (Wide-Angle Beam Propagation Method, WA-BPM), is known as the multistep Padébased technique [1,2,3].We can denote ∂ ∂z with D, and, consequently, ∂ 2 ∂z 2 with D 2 .Eq. S3 can be now viewed as a quadratic equation to be solved for the differential operator D. This yields to the following solution for a first order equation in z: with: Even though it is restricted to forward propagation of the field (z > 0), the above equation is exact in that no paraxiality approximation has been introduced.The radical in Eq.S6 can be evaluated by using a Taylor expansion.The first order of the expansion leads to the standard, paraxial BPM, while higher orders lead to more accurate representations of the propagating field.However, expansion (1,1) (2,2) (3,3) Table S1: Low-order Padé approximants expressed in terms of the operator P defined in Eq.S7.
via Padé approximants is more accurate than Taylor expansion for the same order of terms.This approach leads to the following equation: where N m and D n are polynomials in the operator P , and (m, n) is the order of approximation.Some of their low-order values are reported in Table S1, (1,0) order corresponding to paraxial BPM.Increasing the Padé order allows analyzing larger angles, higher refractive index contrasts, and more complex mode interference both for guided waves and fields propagating in free space.
In this work, in order to balance accuracy and computation time, we made use of the (2, 2) Padé order, corresponding to N 2 = P 2 + P 2 4 and D 2 = 1 + 3P 4 + P 2 16 .
2 CAD model of a single P. strigosum valve CAD models used to simulate the propagation of optical fields through a single P. strigosum valve have been retrieved starting from SEM images of the inner and outer layers of the valve itself.The micrographs have been transformed into binary, bitmap images (see Fig. S1) from which refractive index maps have been derived, assigning the proper values of refractive index to the region occupied by silica and to the environment in which the valve is immersed (air).The refractive index maps have been then extruded and superimposed in order to obtain a 3D CAD model of the valve (overall thickness: 400 nm).The presence of the slits on the external side of the valve and of the bridges occluding the pores of the inner side are not resolved due to the magnification of the starting SEM images.Nevertheless, this does not represent an issue in terms of numerical evaluation of the fields since the linear dimension of these fine features (about 30 nm in width) is one order of magnitude smaller than the optical wavelengths, thus they do not induce detectable diffraction when hit by light.
The direction of propagation of the incoming plane wave was orthogonal to the valve, and the transmitted intensity has been evaluated at different wavelengths, taking into account silica dispersion and absorption.3 Amplitude and phase reconstruction by digital holography The amplitude and phase of the optical wavefront diffused by the object under test can be mathematically retrieved by applying an image reconstruction procedure described in the following.Starting from the frequency spectrum of the acquired interference pattern in off-axis configuration, the first diffraction order is separated from the whole spatial frequency spectrum by a bandwidth filter and shifted to the origin of the plane.As a result, the spectrum of the object field (defined as O(x, y) = |O(x, y)|e iϕ(x,y) , with |O(x, y)| and ϕ(x, y) amplitude and phase, respectively, and x and y cartesian coordinates defining the plane of acquisition of the hologram) is obtained except for a constant [4].
The optical wavefront at different distances from the plane of acquisition can be reconstructed by applying the Fourier formulation of the Fresnel-Kirchhoff diffraction formula [5].The Fresnel-Kirchoff integral, the lens transfer factor, and other operations can be otherwise replaced by operator algebra [6], which allows bypassing the cumbersome integral calculus.In this framework, the propagated field O prop (ξ, η) as a function of the initial field O(x, y) can be expressed as [7]: here F[f (x)] is the Fourier transform of the function f (x), k = 2πn λ (with n refractive index of the medium), ν and µ are spatial frequencies defined as ν = ξ λd and µ = η λd , and d is the reconstruction distance.For digital reconstruction, Eq.S9 is implemented in a discrete form: where N and ∆ are the number of pixels in both directions and pixel dimension, respectively, and m, n, U , V , h and j are integer numbers varying from 0 to N − 1.The discretized Fourier transform is defined as: Intensity and phase distributions of the propagated field can be evaluated by using the following relations: As can be noticed in Eq.S13, the reconstructed phase distribution is obtained by a numerical evaluation of the arctan function, thus its values are restricted in the interval [−π, π], i.e., the phase distribution is wrapped into this range.
To avoid possible ambiguities due to thickness differences greater than λ/2, phase-unwrapping methods have to be generally applied [7].
4 Intensity transmitted by a single valve for λ = 532 nm and λ = 460 nm 5 Transmitted intensity evaluated in XZ plane for different wavelengths In Fig. S3 the spatial distribution of the intensity transmitted by a single P. strigosum valve when invested by a plane wavefront as numerically evaluated by WA-BPM is reported.The transmitted intensity has been calculated in the propagation plane XZ for different values of the incoming wavelength.Silica dispersion and absorption have been taken into account in performing the simulations.When passing from visible to UV-B radiation, the intensity of the transmitted radiation is progressively attenuated and radiation is spatially relocated farther from the valve.

UV-B extinction ratio for living cells in their aquatic environment
In Fig. S4 a transmission micrograph of a single live P. strigosum cell in its growth medium (enriched seawater F/2 medium) is shown when irradiated by UV-B (λ = 280 − 315 nm), together with two intensity profiles evaluated along a segment across a dark area (a) and an area including a brighter region (b), respectively.In both cases the transmitted intensity looks strongly attenuated, with I t /I 0 = 0.23±0.02along segment a and I t /I 0 = 0.64±0.02along segment b (I t and I 0 standing for average transmitted and incident intensity, respectively).

Figure S1 :
Figure S1: Binary images of the outer (a) and inner (b) layer of a P. strigosum valve as retrieved from SEM micrographs.

Figure S4 :
Figure S4: Intensity profiles evaluated along two different segments across a live P. strigosum cell in its growth medium (enriched seawater F/2 medium) when illuminated by UV-B radiation (λ = 280 − 315 nm): dark area (a); area including a brighter region (b).